No, It is currently not possible to upgrade YubiKey Firmware. To prevent attacks on the YubiKey which might compromise its security, the YubiKey does not permit its firmware to be accessed or altered.

Yubico is dedicated to providing a long-term two-factor authentication solution, we want your YubiKey to remain useful for the full extent of it’s lifetime. When we do release new firmware, we ensure the new YubiKey will function the same as with older versions, so there is no need to purchase new YubiKeys to ensure compatibility.